Average Error: 59.4 → 0.8
Time: 3.0m
Precision: 64
Internal Precision: 128
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[\log \left(\frac{\sqrt[3]{e^{\frac{f \cdot \pi}{4}} + e^{\left(-f\right) \cdot \frac{\pi}{4}}}}{\sqrt[3]{(\pi \cdot \left((\left(\left(f \cdot f\right) \cdot \frac{1}{192}\right) \cdot \left(f \cdot \left(\pi \cdot \pi\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left({f}^{5} \cdot \left({\pi}^{5} \cdot \frac{1}{61440}\right)\right))_*}}\right) \cdot \frac{-4}{\pi} + \left(\log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{\left(-f\right) \cdot \frac{\pi}{4}}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{\left(-f\right) \cdot \frac{\pi}{4}}}}{\sqrt[3]{\left(\left({\pi}^{5} \cdot {f}^{5}\right) \cdot \frac{1}{61440} + \frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right) + \frac{1}{2} \cdot \left(f \cdot \pi\right)} \cdot \sqrt[3]{\left(\left({\pi}^{5} \cdot {f}^{5}\right) \cdot \frac{1}{61440} + \frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right) + \frac{1}{2} \cdot \left(f \cdot \pi\right)}}\right) \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right) \cdot \left(-\sqrt{\frac{4}{\pi}}\right)\]

Error

Bits error versus f

Derivation

  1. Initial program 59.4

    \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]

  2. Taylor expanded around 0 0.9

    \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right)\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt1.2

    \[\leadsto -\color{blue}{\left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right)} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)\]

  5. Applied associate-*l*0.8

    \[\leadsto -\color{blue}{\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)\right)}\]

  6. Simplified0.8

    \[\leadsto -\color{blue}{\sqrt{\frac{4}{\pi}}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right)\right)\]
  7. Using strategy rm

  8. Applied add-cube-cbrt0.8

    \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\left(\sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)} \cdot \sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right) \cdot \sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}}\right)\right)\]

  9. Applied add-cube-cbrt0.8

    \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \left(\frac{\color{blue}{\left(\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}}{\left(\sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)} \cdot \sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}\right) \cdot \sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right)\right)\]

  10. Applied times-frac0.8

    \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \log \color{blue}{\left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)} \cdot \sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}} \cdot \frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right)}\right)\]

  11. Applied log-prod0.9

    \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \left(\sqrt{\frac{1}{\frac{\pi}{4}}} \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)} \cdot \sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right) + \log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right)\right)}\right)\]

  12. Applied distribute-rgt-in0.9

    \[\leadsto -\sqrt{\frac{4}{\pi}} \cdot \color{blue}{\left(\log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)} \cdot \sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right) \cdot \sqrt{\frac{1}{\frac{\pi}{4}}} + \log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right) \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right)}\]

  13. Applied distribute-rgt-in0.9

    \[\leadsto -\color{blue}{\left(\left(\log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)} \cdot \sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right) \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right) \cdot \sqrt{\frac{4}{\pi}} + \left(\log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right) \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right) \cdot \sqrt{\frac{4}{\pi}}\right)}\]

  14. Simplified0.8

    \[\leadsto -\left(\left(\log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}}{\sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)} \cdot \sqrt[3]{\frac{1}{2} \cdot \left(f \cdot \pi\right) + \left(\frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + \frac{1}{61440} \cdot \left({f}^{5} \cdot {\pi}^{5}\right)\right)}}\right) \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right) \cdot \sqrt{\frac{4}{\pi}} + \color{blue}{\frac{4}{\pi} \cdot \log \left(\frac{\sqrt[3]{e^{\frac{f \cdot \pi}{4}} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}}{\sqrt[3]{(\pi \cdot \left((\left(\left(f \cdot f\right) \cdot \frac{1}{192}\right) \cdot \left(f \cdot \left(\pi \cdot \pi\right)\right) + \left(\frac{1}{2} \cdot f\right))_*\right) + \left({f}^{5} \cdot \left(\frac{1}{61440} \cdot {\pi}^{5}\right)\right))_*}}\right)}\right)\]

  15. Final simplification0.8

    \[\leadsto \log \left(\frac{\sqrt[3]{e^{\frac{f \cdot \pi}{4}} + e^{\left(-f\right) \cdot \frac{\pi}{4}}}}{\sqrt[3]{(\pi \cdot \left((\left(\left(f \cdot f\right) \cdot \frac{1}{192}\right) \cdot \left(f \cdot \left(\pi \cdot \pi\right)\right) + \left(f \cdot \frac{1}{2}\right))_*\right) + \left({f}^{5} \cdot \left({\pi}^{5} \cdot \frac{1}{61440}\right)\right))_*}}\right) \cdot \frac{-4}{\pi} + \left(\log \left(\frac{\sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{\left(-f\right) \cdot \frac{\pi}{4}}} \cdot \sqrt[3]{e^{\frac{\pi}{4} \cdot f} + e^{\left(-f\right) \cdot \frac{\pi}{4}}}}{\sqrt[3]{\left(\left({\pi}^{5} \cdot {f}^{5}\right) \cdot \frac{1}{61440} + \frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right) + \frac{1}{2} \cdot \left(f \cdot \pi\right)} \cdot \sqrt[3]{\left(\left({\pi}^{5} \cdot {f}^{5}\right) \cdot \frac{1}{61440} + \frac{1}{192} \cdot \left({f}^{3} \cdot {\pi}^{3}\right)\right) + \frac{1}{2} \cdot \left(f \cdot \pi\right)}}\right) \cdot \sqrt{\frac{1}{\frac{\pi}{4}}}\right) \cdot \left(-\sqrt{\frac{4}{\pi}}\right)\]

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed 2018349 +o rules:numerics
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  (- (* (/ 1 (/ PI 4)) (log (/ (+ (exp (* (/ PI 4) f)) (exp (- (* (/ PI 4) f)))) (- (exp (* (/ PI 4) f)) (exp (- (* (/ PI 4) f)))))))))